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Quadratic  Involutions  on  the  Plane 
Rational  Quartic 


DISSERTATION 

Submitted  to  the  Board  of  University  Studies  of  the  Johns  Hopkins  University 

in  conformity  with  the  requirements  for  the  degree  of 

Doctor  of  Philosophy 


BY 

THOMAS  BRYCE  ASHCRAFT 
1911 


V 


,r»R 


I 


Of    tHf  \ 


rAU'r^^'*''*'   -^ 


.«"' 


Press  of  the 

Mail  Publishing  Company 

Waterville,  Maine 


Quadratic  Involutions  on  the  Plane 
Rational  Quartic 


DISSERTATION 

Submitted  to  the  Board  of  University  Studies  of  the  Johns  Hopkins  University 

in  conformity  with  the  requirements  for  the  degree  of 

Doctor  of  Philosophy 


BY 

THOMAS  BRYCE  ASHCRAFT 
1911    V      ■ 


Press  of  the 

Mail  Publishing  Company 

Waterville,  Maine 


A 


^'^■(^1 


,< 


Quadratic  Involutions  on   the  Plane  Rational  Quartic. 

By  T.  B.  Ashcraft. 


§  I .  The  General  Theory  of  Involution  Curves  of  a  Plane  Rational  Curve  of  Order  n. 

Let  R"  denote  a  plane  rational  curve  of  order  n,  and  let  it  be  given  by 
the  equation 

Xo  =  Oo<"  +  a,^""'  +  ajr'  +  — -  +  a„ 
(i)  x^^bjf+b.t"-^  +bjr-'+—-  +Z>„ 

x^  =  c^  +c,^-'  +cjr'  +— -  +c. 

If  we  join  the  parameters  t^  and  /j  by  a  line,  where  /,  and  /j  are  in  an  involu- 
tion of  the  form 

(2)  A  t^,  +  5(/,  +  /j)  +  C  =  o, 

we  shall  show  that  the  locus  of  this  line  is  a  rational  curve  of  class  n-i  which 
touches  R'  2){n-2)  times  and  meets  it  in  2(n-2){n-T,)  other  points.  This  class 
curve  will  be  called  an  involution  curve,  and  will  be  denoted  by  r*"'. 

Cut  the  curve  R"  by  any  line 

(3)  (fx)  =fo»o  +  fi»i +^2^2=0 
and  we  have 

(4)  («„$)  r  +  (« .0  /-  + +  {a J)  =  o. 

For  convenience  suppose  we  choose  the  involution  with  o  and  00  as  double 
points.  Then  /  and  -t  must  satisfy  the  last  equation,  and  we  have  for  n  even, 
say  n  =  2  m, 


263424 


2       T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

(5)  («oO  t"'  +  («iO  ^•»-'  +---  +  (a„f)  =  o, 

(6)  («of )  f ""—  (« .f )  /'"-'  +  -—  +  (««0  =  o. 
Whence  by  addition  and  then  subtraction  we  get 

(7)  («oO  i"'  +  («2f)  t""-'  +  -----  +   («„f)  =  o, 

(8)  («.f)  f^-'  +  .(«3f)  ^^"-^  +  —  +  («„-xf)  =  o. 

Since  only  even  powers  occur  we  can  divide  the  exponent  by  2  and  write 

(9)  («of)  i"  +  («20  t"-'  +---  +  (««f)   =  o, 

(10)  («.f)  r-'  +  («30  /«-'  +  -----  +  (a„-.e)  =  o. 

EHminating  $„,  f  „  f  2  from  equations  (3),  (9)  and  (10),  we  have  the   locus  re- 
quired in  determinant  form 

(11)  /o(r)    ,   /.(/")     ,   /,(r) 

Or  written  parametrically  its  equation  is 

and  since  n  =  2w  we  have  as  the  representation  of  the  involution  curve 

(12)  f,  =  F,(r'), 

which  is  a  rational  class  curve  of  order  n-i.     Similar  argument  holds  for  n 
odd. 

The  point  to  be  emphasized  is  that  the  parameter  may  be  replaced  by  a 
new  one  which  reduces  the  degree  by  one  half;  that  is  a  T  replaces  a  quadratic 
in  t.  This  new  parameter  may  be  chosen  in  a  triple  infinity  of  ways  depend- 
ing on  the  ratios  of  a,  ^,  v,  5  in  a  transformation  of  the  form 

ctt   +   ?> 

1  =    


vi  +  S      . 

If  the  double  points  of  the  involution  are  given  by  (a  ty,  then  we  choose  any 
two  quadratics  apolar  to  (a  t)',  say  (a/)*  and  (^  /)';  then  any  convenient  mem- 
ber of  the  pencil  («  t)'  +  X  (p  i)'  will  serve  as  a  new  parameter  T. 

To  find  the  number  of  contacts  of  the  r"''  with  the  R",  we  shall  consider 
first  the  R*  and  its  involution  cubic  r^.  The  R"*  is  of  class  six  so  there  are  18 
common  lines.  There  are  three  ways  in  which  we  may  have  common  lines. 
A  line  meets  the  curve  in  four  points,  /„  ^2,  t},  tt.  Let  ^,  and  /j  come  together 
so  that  line  is  a  tangent.      We  have  common  lines  when 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic     3 

i)      Points  ty  and  t^  are  a  pair  of  the  involution, 

2)  Points  t^  and  /<  are  a  pair  of  the  involution, 

3)  Points  /,  and  t^  are  a  pair  of  the  involution. 

Case  i)  can  happen  twice,  viz.  when  the  line  cuts  out  either  of  the  double 
points  of  the  involution.     This  accounts  for  two  common  lines. 

In  case  2)  a  tangent  at  t  meets  the  curve  again,  say  at  t^.  For  a  given 
t  there  are  two  /,'s,  and  for  a  given  ^,  there  are  4  i's,  since  there  are  four  tangents 
from  a  point  on  the  curve,  the  curve  being  of  class  six.  The  relation  connecting 
t  and   ti  is 

/.  (<*)  t*  +  f,  (t*)  t,  +  f,  (t^)   =  o. 

The  condition  that  the  roots /,  be  in  an  involution  is  of  the  fourth  degree 
in  t,  which  means  four  common  lines  for  case  2).  Case  3)  must  contain  all 
the  other  common  lines,  that  is  twelve.  This  case  happens  when  /j  and  tj 
are  a  pair  of  the  involution,  but  ^2  and  ^3  are  as  well  a  pair  of  the  involution; 
therefore  the  twelve  common  lines  are  six  repeated.  In  other  words  the  R* 
has  six  contacts  with  the  r^. 

This  is  easily  extended  to  the  general  case  of  R".  The  R"  is  of  class 
2(«-i),  so  the  R"  and  the  r"''  have  2(«-i)^  common  lines.-  There  will  always 
be  two  of  these  accounted  for  in  case  i),  corresponding  to  the  double  points 
of  the  involution. 

For  case  2)  the  equation  connecting  a  point  of  tangency  /  and  a  point  of 
intersection  of  the  tangent  at  /,  is  of  degree  2W-4  in  t  and  n-2  in  /„  and  is  of  the 
form 

/  (/"-^  /i"-»)  =0. 

For  two  values  of  ti  to  be  in  a  given  involution  is  a  condition  of  degree 
n-2,  in  the  coefficients  of  ^,  and  hence  of  degree  2(w-2)(«-3)  in  t;  this  then  is 
the  number  of  common  lines  for  case  2).  Subtracting  the  common  lines  for 
case  i)  and  case  2)  from  the  total  number  we  have  for  case  3)  2(«-i)» 
— 2(n-2)(n-3) — 2  =  6n-i2.  But  since  these  pair  ofif  we  have  in  general  2>n-6 
contacts  of  R"  and  r""'. 

The  order  of  r""'  is  2  «  -  4,  so  the  R"  and  its  involution  curve  intersect  in 
2  n  (n  -  2)  points.  The  contacts  count  for  6  n  -  12  intersections,  so  there  are 
2(«-2)(«-3)    remaining    intersections. 

If  the  parameters  of  a  node  of  R"  are  in  the  involution,  then  the  node  is  a 
factor  of  the  involution  curve,  and  the  remaining  factor  is  an  r"''.  This 
part  of  the  locus  being  a  double  point  of  the  R"  will  count  for  two  contacts; 


4      T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

hence  the  remaining  part,  r"'',  will  have  only  3  «  -  8  contacts.  If  n-T,  nodes  are 
made  a  part  of  the  locus  then  we  always  get  for  the  remaining  part  of  the  locus 
an  r'  with  n  contacts  or  full  contact,  that  is  a  conic  all  of  whose  intersections 
are  contacts.  Since  two  sets  of  the  involution  determine  the  involution 
it  is  a  condition  on  the  R"  for  n-3  nodes  to  be  in  the  involution  for  n  greater 
than    5.* 

We  shall  now  consider  the  R^  and  find  the  involution  conic.  We  found 
that  there  are  in  general  3W-6  contacts,  so  in  this  case  we  get  three  contacts 
and  no  extra  intersections.  Let  the  double  points  of  the  involution  be  o 
and  00  ,  and  let  the  reference  triangle  be  the  tangents  at  the  double  points  and 
their  join.     Then  the  equation  of  R^   may  be 


(I) 


Xo  =  aat^+bat' 
Xi  =Cit  +di 
X,  =  b,t'  +cd. 


If  o  and  00  are  the  double  points  of  the  involution  it  is  of  the  form 

<2)  ^,+^2=0, 

and  the  line  whose  locus  is  the  involution  conic  will  join  t  and  -t  of  the  R'. 
Such  a  line  is  given  by 

=  0. 


^0 

> 

X, 

X, 

a,t^+h,t* 

» 

c^t  +i, 

b,t'  +c,t 

-a^^  +  b^t' 

> 

-c,t  +  di 

b,t'-  c,t 

This  determinant  is  readily  seen  to  reduce  to  the  following: 


(4) 

Xq          ,       Xi       ,       Xi 

bot'     ,     d,     ,     b,t 
aj.'     ,     Ci     ,     Cj 

which  is  parametrically 

(5) 

$0=  — b^Cit'  -\-Cidi 
f  ,=  ajb^t* — b^^f 
^2=  (^0^1 — a4i)  t'. 

*In  a  Desargues  Configuration  B  there  is  a  sextic  with  nodes  at  the  ten  points  of  the  Configura- 
tion. Any  three  nodes  on  a  line  would  be  in  an  involution,  hence  we  could  get  ten  conies  having  full 
contact  with  the  sextic. 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic       5 

It  is  seen  that  only  even  powers  of  t  occur  so  we  replace  the  parameter  t' 
by  a  new  one,  t'  say.  For  convenience  we  drop  the  primes  and  write  the 
equation   in   the   form 

f  0  =  — biCit  +  Cidi 

(6)  $i=a„b2t' — 60^2^ 
$2=  {b^Ci — a^di)  t 

This  is  the  involution  conic  in  line  form  but  we  want  the  point  form.  We 
have 

^0  =  ^0^2  ((^odi—boCi)  t' 

(7)  ^1  =  Cidi  {aod^—boC^) 

X2  =  — aJ)2Cit'  +  20062^2^^1^ — bgCldi. 

Now  in  order  to  get  the  intersections  of  this  involution  conic  with  the  R' 
we  must  eliminate  the  parameter  from  the  equation  of  the  conic  and  thus  get 
an  equation  of  the  second  degree  in  x.  If  we  then  substitute  for  the  x^'s  their 
values  in  the  equation  of  the  R^  we  obtain  a  sextic  in  t  which  will  give  the 
intersections  of  the  two  curves. 

Eliminating  /  from   (7)   w  e  get 

bi'Ci'Xg'  +  b^c^x'  +  {a^d'  —  2a^gCyd^  +  b^c^)  x* 

(8)  +  2  {aJbaCidi  —  bg'c^c^  XyXi  +  2  {aab^c^d^  —  60^2^1*)  ^0*2 

+  2  (  bobiCiCi  —  2  aobiCidi)  ^o^i    =   O- 

If  we  now  substitute  for  the  x/s  their  values  in  equation  (i)  we  get 

ao'b^'Ci't^  +  2  a^'b^'c^dfi  +  (ao'b^'di'  —  2ag'b2C,C2di)  t* 

(9)  —  (2  ao'b^Ctdi'  —  2  aobobiCiCjdi  )  t^   +  {a^'c^'d' —  2  aobJbiC^d^')  f 

+  2  ajy^f^d^t  +  b^c^d*  =   o. 

This  sextic  is  seen  to  be  the  square  of  the  cubic 

(10)  aJb^Cit^  +  ajb^dif  —  agCidit  —  60^2^1  =  o, 

which  gives  the  parameters  of  the  three  points  of  contact  of  the  R'  and  its 
involution  conic. 

We  shall  now  prove  that  the  points  of  contact  of  an  R^  and  its  involution 
conic  are  given  by  the  Jacobian  of  the  cubic  giving  the  parameters  of  the  three 
flexes  of  R^  and  the  quadratic  which  gives  the  double  points  of  the  involution. 


6       T.  B.  A  slier  aft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

We  shall  consider  the  R^  given  by  equation  (i),  and  the  involution  whose 
double  points  are  o  and  oo .  The  cubic  giving  the  flexes  is  the  fundamental 
cubic,  that  is,  the  unique  cubic  apolar  to  each  of  the  three  binary  cubics  in  (i). 

Calculating  that  cubic  in  the  usual  way  we  have 

(ii)  a^biCtt^  +  2>ciob2dit'  +  scoCa^i^  +  b^Cidi  =  o. 

The  quadratic  giving  the  double  points  under  consideration  is 

(12)  t  =  o. 

The  Jacobian  of  (11)  and  (12)  is 

(13)  agbiCit^  +  ajb^dit""  —  a^Cidit  —  5o^2^i  =  o, 

and  is  just  the  same  cubic  as  (10),  and  the  theorem  is  proved. 

We  shall  now  consider  R'*  and  its  involution  cubic  r^.  The  number  of 
contacts  we  found  to  be  in  general  3«-6  which  is  just  the  number  of  flexes  of 
an  R".  Since  the  Jacobian  of  the  flex  cubic  and  the  quadratic  of  the  double 
points  of  the  involution  gave  the  contacts  of  R^  and  r^  it  seems  natural  to  look 
for  some  such  relation  in  the  case  of  R'*.  The  Jacobian  of  the  flex  equation 
in  general  and  the  quadratic  giving  the  roots  of  the  involution  will  always  be 
of  the  right  degree  in  t,  3W-6,  to  give  the  points  of  contact  of  R"  and  r""'. 
But  in  the  case  of  R'*  we  find  the  degree  in  the  coefficients  not  the  same  as  those 
of  the  contact  equation.  We  shall  find  by  a  symbolic  method  the  degree  of 
the  contact  equation  in  the  coefficients  of  the  fundamental  involution,  as  well 
as  in  the  coefficients  of  the  quadratic  of  the  involution. 

Suppose  the  fundamental  involution  of  R*  is  given  by 

(i)     (aty  + 1  i^ty  =  o. 

Let  the  double  points  of  the  involution  be  (Qty  =  o,  and  let  one  set  of  the 
involution  /,  and  t^  be  given  by  (at)'.     Let  the  line  on  /i  and  ^2  meet  the  R* 
again  at  T,  and  Tj. 
Since  every  line  section  is  apolar  to  the  fundamental  involution  we  have 

(2)  \aa\  (aT,)  (aTj)  =  o     and 

(3)  M  (^TO  (^T,)    =  o. 
Eliminating  Tj  from  (2)  and  (3)  we  get 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic      7 

(4)  |«^||«aMM.'|(«T,)  (PT.)  =0. 

Since  every  set  of  the  involution  is  apolar  to  (Qt)',  we  have, 

(5)  \aQ\'   =  o. 

Again,  since  when  /,  or  /^  is  T,,  there  is  a  contact,  we  have 

(6)  (aT,)»=o. 

Solving  for  the  a's  in  (5)  and  (6)  we  find  them  to  be  of  the  first  degree  in 
the  Q's  and  of  the  second  degree  in  the  T's.  If  these  values  of  the  a's  are  put 
in  (4)  we  get  an  equation  of  the  first  degree  in  the  determinants  of  the  funda- 
mental involution,  of  the  first  degree  in  the  Q's,  and  of  the  sixth  degree  in  T. 
This  is  the  contact  equation. 

The  flex  sextic  is  the  first  transvectant  of  the  fundamental  involution  and 
is  of  the  first  degree  in  the  determinants  of  the  fundamental  involution.  The 
Jacobian  of  the  flex  sextic  F  and  the  quadratic  Q  giving  the  roots  of  the  in- 
volution is  a  sextic  Ji  which  is  of  the  first  degree  in  the  determinants  of  the 
fundamental  involution,  but  only  of  the  first  degree  in  the  Q's.  Taking  the 
Jacobian  of  J,  and  Q  we  get  a  sextic  J 2  which  is  of  the  second  degree  in  the 
Q's  and  the  first  degree  in  the  determinants  of  the  fundamental  involution. 

Now  we  propose  to  show  that  K,  the  sextic  giving  the  points  of  contact 
of  R*  and  r^,  can  be  built  from  F,  Q,  J 2,  A,  and  q,  where  F,  Q,  J 2  have  the 
meaning  just  given,  and  where  A  is  the  discriminant  of  Q,  and  q  is  the  third 
transvectant  of  the  two  members  of  the  fundamental  involution.  The  possi- 
ble combinations  that  are  of  the  same  degree  as  K  are  easily  seen.  We  shall 
show  that 

K  =  X  A  ¥  +^]2+vgQ\ 

where  X,  iJi,  v  are  constants  to  be  determined.  Let  the  R*  be  referred  to  two 
flex  tangents  and  the  line  joining  these  flexes  whose  parameters  are  o  and  00 . 
Its  parametric  equation  will  be 

Xq  =  at*  +  bt^  -It  c1^ 
(l)  x^  =  bt^  +  ct'  +  dt 

Xj  =  cf  +  dt  +  e 


8       T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 


We  shall  choose  the  involution  whose  sets  are  t  and  -/,  hence  whose  double 
points  are  given  by 


(2) 


Q   -2/ 


o. 


Since  we  are  not  interested  in  the  equation  of  r^,  we  proceed  to  find  the 
equation  giving  the  points  of  contact  with  the  R*.  Calculating  the  funda- 
mental involution  of  R*,  that  is  two  quartics  which  are  apolar  to  the  three 
binary  quartics  in  (i),  we  get 

(3)  bet*  —  6b  ef  —  4  c  e  /  =  o,     and 

(4)  ^act^+6adf  —  c  d  =  o. 

The  polarized  form  of  (3)  and  (4),  that  is  where  S  refers  to  f,,  t^,  t,,  t^,  is 

(5)  be  St  —  beSi  —  ceSi=o     and 

(6)  aeS3  +  adS2  —  ed=o, 

which  is  known  to  be  the  condition  that  four  points  lie  on  a  line.  Now  let  two 
of  the  t's,  say  t^  and  /j  be  equal  to  /,  and  let  a"  refer  to  t^  and  1^     Then  (5)  and 

(6)  become 

(7)  (bet'  —  b  e)  0  2  —  (2  6e^+ce)<T,  —  bet'  —  2  c  e  t  =  o, 

(8)  {2act  +  ab)<s2  +  {act'  +  2adt)  o^  +  a  d  t'  —  c  d  =  0. 
Taking   also   the   equation 


(9) 


TJ,  +  X' 


o, 


and  eliminating  the  j's  from  (7),  (8),  (9)  we  have 
(10) 


bet'  —  be         ,  —  2b  e  t  —  e  e  ,  —  bet'  —  2  c  e  t 
2  a  c  t  +  a  d  ,  a  c  t'  +  2  a  d  t ,  a  d  t'  —  c  d 


=  o. 


This  is  an  equation  which  is  obviously  the  equation  giving  the  para- 
meters of  the  six  flexes  when  x  is  t,  and  giving  the  parameters  of  the  six  points 
of  contact  of  R*  and  r^  when  t  is  -t.  Putting  x  =  t  and  developing  (10) 
we  get 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic    9 

(11)  F  =  abc't:^  +  sabcdt^  +  6  ab  c  e  t*  +  (8  ac' e  —  b  c*  d)  t^ 

■\-6acdei'+2^bcdet-\-c'de=o. 

If  T  =  —  t ,  (10)  becomes 

(12)  K  ^  abc'f"  +  abcdt^  +  2abcet*  +  be"  dt^ 

+  2  acdet'  +  b  cde  t  +  c'  d  e  =  o. 

The  Jacobian  of  F  and  Q  is  v 

(13)  J,  ^  ab  c'  f  +  2  ab  c  dt^  +2abcet*  — 2acdet' 

—  2  b  c  d  e  t  —  c'  d  e  =0. 

The  Jacobian  of  J ,  and  Q  is 

(14)  J2  =  ^ab  c'  ^  +  ^ab  cdt^  +  2  ab  c  e  t*  +  2  a  c  d  e  f 

+  /\.b  c  d  e  t  +  2)  C  d  e  =  0. 

The  quadratic  ^  is  the  third  transvectant   of    the  members    of    the    funda- 
mental involution. 

Taking  the  third  transvectant  of  (3)  and  (4)  we  have 

(15)  ^  =  T,abcet'  +  {2  ac'  e  +  b  c'  d)  t  +  ;^  a  c  d  e  =0. 
Forming  the  product  of  ^  and  Q',  we  get 

(16)  fQ'  =  J  2abcet*  +  {8ac'  e  +  ^bc'  d)  t^  +  1  2  a  c  d  e  t'  =0. 
The  discriminant  of  Q  is 

(17)  A   =  I. 
Writing  down  now 

we  find  that  K  is  given  for 

.   >^  =  —  1/5,  V-  =  2/5.  >   =  i/5- 
Or  to  avoid  fractions  we  have  finally 

(18)  5K  =i.Q'  +2j,-AF. 


10     T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

If  two  nodes  are  in  the  involution,  these  two  nodes  are  a  factor  of  the  r' 
and  the  remaining  factor  is  some  other  point.  We  shall  show  that  the  re- 
maining factor  of  r^  is  the  third  node. 

Let  the  parameters  of  one  node  be  given  by  /'  +  a,  a  second  by  /*  +  h, 
and  the  third  by  a  general  quadratic  c^t'  +  Cit  +  Cj.  The  R*  referred  to 
its  nodes  has  the  equation 

x^  =  {f  +  a)  (co  t"  +  c^t  +  Cj) 
(19)  X,  =  [f  +  b)  (co  t'  +  c^t  +  cj 

«2  =  (/'  +  a )  {f  +  b) 

The  sets  of  the  involution  are,  if  two  nodes  are  in  it,  t  and  -/.       The  equation 
of  a  line  joining  t  and  -/  is  given  by  the  determinant 


Xa 


X. 


X, 


=  o. 


{f  +  a)  {cJ."  +  c,^  +  Cj)  ,  (/'  +  b)  (cot'  +c,t+C2)  ,  (f  +  a)  (f  +  b) 
(t'+a)  Ic^t'—c^t+c,)  ,  it'  +b)  Icot'—c^t+c^)  ,  (f  +a)  (f  +b) 

If  now  we  take  the  sum  of  the  second  and  third  rows  for  a  new  second 
row,  and  their  difference  for  a  new  third  row,  and  remove  the  factor  Ci^  from 
the  third  row,  we  have 


(20) 


x„ 


(f  +  a)  (co  t'  +  c,) 
t'  +  a 


it'  +  b)  (Co  t'  +  c)  ,'  (f  +  a)  It'  +b) 
t'  +  b  .  o 


=  0. 


Replacing  f  by  T  and  expressing  this  equation  in  terms  of  f  ,'s  we  have  after 
removing  common  factors 

f  0  =  -  (T  +  6) 
(21)  e.  =  T  +  a 

f  J  =  o 


which  shows  on  the  face  of  it  the  other  node  to  be  the  rest  of  r». 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic     II 


§  2.     Involutions  Determined  by  Two  Double  Lines  of  the  Plane 

Rational  Quartic. 


If  lines  are  drawn  on  the  meet  of  any  two  double  lines  of  the  rational 
quartic,  we  obtain  a  quadratic  involution.  That  is  to  say,  while  such  a  line 
meets  the  curve  in  four  points  the  parameters  pair  off,  /j  and  t^  say. 

By  choosing  o  and  oo  as  the  points  of  contact  of  one  double  tangent  we 
may  write  the  curve 

(i)  x^  =  {aif  +  2  bit  +  c^y 

X2  =  (oj  r  +  2  62  /  +  C2)'  . 

Any  line  on  the  meet  of  Xg  and  Xi  is  of  the  form  x^  —  X'  a;,  =  o,  or 

(2)  /^t'  —  X'  {aj'  +  2b  it  +  CiY  =  o 

which  breaks  into  factors 

(3)  [  2  /  —  X  (a,  f  +  2  Z>,  /  +  c,)]  [  2  /  +  X  (  a,  /'  +  2bit  +  c,)]  =  o. 

If  ti  is  a  root  of  the  first  factor,  then 

(4)  X  =  


a,  /'  +  2bit  +  Ci 
Substituting  this  value  we  have,  after  removing  the  factor  t  -/,  , 
(5)         -^(0,1)  =  ai^^2  — Ci  =  o, 


12     T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

and  this  is  the  quadratic  involution  from  the  meet  of  x^  and  Xi. 

We  shall  denote  the  double  lines  by  o,  i,  2,  3,  and  /„;^-)  will  denote  the 
quadratic  involution  obtained  by  drawing  lines  on  the  meet  of  any  two  double 
lines  *  and  j. 

We  shall  show  that  the  double  points  of  I  ,0.,)  are  given  by  the  points  of 
contact  of  the  two  remaining  tangents  from  the  meet  of  the  double  lines  o  and  i. 

The  double  points  of  /(o,i)  are  given  by 

(6)  a,  ^  —  c,  =0. 

It  is  easily  seen  that  the  Jacobian  of  the  quadratics  which  give  the  points 
of  contact  of  the  double  lines  give  the  points  of  contact  of  the  two  remaining 
tangents  from  their  meet;  for  the  Jacobian  of  two  squared  quadratics  is  the 
product  of  the  quadratics  and  their  Jacobian.  Forming  the  Jacobian  of  the 
double  lines  o  and  I  we  have 

(7)  a,/'  — c,  =  o    , 

which  gives  precisely  the  roots  of  /(o.d- 

We  shall  next  prove  that  the  points  of  contact  of  the  two  tangents  that  may 
be  drawn  to  the  quartic  from  the  meet  of  any  two  double  lines  are  on  a  line 
through  the  meet  of  the  other  two  double  lines.  Having  proved  that  the 
roots  of  the  involution  I  (o,,)  are  the  points  of  contact  of  the  other  two  tangents 
from  the  meet  of  o  and  i,  we  have  only  to  show  that  the  points  of  contact 
of  tangents  from  the  meet  of  the  double  lines  2  and  3  are  in  the  involution 
7(0.1),  that  is  to  say,  that  the  roots  of  I^2,))  are  in  /(o,i)- 

Not  having  the  equation  of  the  double  line  3,  we  make  use  of  the  well- 
known  fact  that  the  three  catalectic  sets  of  the  fundamental  involution  give 
the  three  sets  of  two  tangents  from  the  meets  of  double  lines,  such  as  from  o 
and  I,  and  2  and  3.  The  fundamental  involution  of  the  quartic  given  by  (i) 
takes  the  form 

(8)  |a,  bi  b^c.lt*  +  I  a,  6,  c,'  \  t>  +  {b^c,  c,"  |  t 

+  X  (  I  o,  &,  0/  I  /3  +   I  62C2  a,'  I  ^  +   I  a,  ft,  6jCj|)  =0, 


where  |  a,  Z),  63  C2  |  denotes  the  determinant 


a,  bi ,  a^  62 
6,c, ,  62  C2 


and    I  a,  &,  c*\  is 


a,  bi  ,  a^  62 


and  so  on. 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic     13 
Writing  down  the  g^  of  (8)  we  have 


(9) 


a,  6,^2*  I  +  X  I  a,  6,a/I,  o  ,  j&jCjC,"  1 +>>  IftzCi^i' I 

o  ,|62C2C,»|  +  Xl&jCja.'l,     X|a,  6,6,^2! 


=  0. 


This  is  a  cubic  in  X  whose  roots  are  at  once  found  to  be 

c,'  c,'  {biC,  —  b,c,y 


a,*  at'  (o,  &2  —  a^biY 

Substituting  X  = in  (8)  we  get,  after  removing  the  factor 

a' 
biia^c^  —  a^Ci), 

(10)  o,"  6j  t*  +  a,  (a,  Cj  +  ^2  c, )  /3  —  c,  (a,  c^  +  aj  c, )  / —  62  c,'  =  o. 
This  factors  into 

(11)  [a^f  —  Ci]  [  fli  62  ^'  +  ( <*i  ^2  +  ^2  c, )  ^  +  ^2  ^1 1  =  o. 

the  first  factor  giving  the  points  of  contact  of  tangents  from  (o,  i)  and  the 
second  factor  giving  the  points  of  contact  of  the  two  tangents  from  (2,  3), 
where  (o,  i)  means  the  meet  of  o  and  I,  and  so  on.  We  have  now  to  show  that 
the  roots  of 

(12)  a,  &2  f  +  (a,  Cj  +  O2  c, )  /  +  62^1  =0 

are  in  /,o_,).  The  only  condition  necessary  is  that  the  product  of  the  roots 
shall  be  c  '/a,,  and  this  is  obviously  so  in  the  case  of  (12). 

We  get  7(2,3)  by  polarizing  (12).     Thus 

(13)  -^(2,3)  =  2  a,  Z)j  /i  ^2  +  (  ^1  C2  +  a^  c^)  (ti  +  t^)  +2  bj  c,  =  o. 

It  is  then  also  readily  seen  that  the  roots  of  /,o.i)  are  a  set  of  7,2,3).  The 
fact  that  the  double  points  of  7,0,,)  are  in  7,2,3,  and  also  that  the  double  points 
of  7(2.3,  are  a  set  of  7,o.i)  says  the  two  involutions  are  commutative,  that  is 
•^(0,1)  -^(2,3)  =  -^(2.3)  Ao.D-  Thus  there  are  a  single  infinity  of  four-points  on 
the  curve  for  which  (o,  1)  and  (2,3)  are  diagonal  points.  If  we  allow  this  four- 
point  to  run  around  the  curve,  (o,  i)  and  (2,  3)  will  be  fixed  diagonal  points 


14     T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

and  the  third  diagonal  point  will  have  a  locus.  There  will  be  three  such  loci 
corresponding  to  the  three  ways  in  which  we  may  pair  off  the  double  lines. 
This  question  will  be  considered  in  a  subsequent  paragraph  of  this  paper. 

We  have  proved  that  the  points  of  contact  of  the  two  remaining  tangents 
from  the  meet  of  any  two  double  lines  lie  on  a  line  through  the  meet  of  the  other 
two.  There  are  then  six  such  lines  and  we  shall  prove  they  are  on  four 
points.  We  shall  denote  by  L,o,,)  the  line  on  the  points  of  contact  of  tangents 
from  (o,  i),  that  is  from  the  meet  of  the  double  lines  o  and  i.  The  other 
five  lines  are  similarly  named.  It  is  obvious  that  if  three  lines  are  on  a  point 
they  must  be  such  as  L(o,i)..  L,(o,2)  L(o,3)-  To  get  these  three  lines  we  need  the 
double  points  of  /(o,i),  /(0.2)  and  /(o,3).     We  have  found  the  roots  of  /(o,i)tobe 

(14)  a^t"  —  c,  =  o. 

From  symmetry  the  roots  of  7, 0,2)  are 

(15)  a^f  —  c,  =  o. 

Now  finding  the  roots  of  7,  ,,2)  and  again  making  use  of  the  catalecttc  sets 
we  find  the  roots  of  7(o,3)  to  be 

(16)  (ai&2  —  Ojft,)/'  —  {biC^  — b^c\)  =  o. 

The  roots  in  these  three  involutions  are  given  by  equations  with  no  middle 
term;  hence  the  three  lines  wanted  are  in  each  case  on  parameters  t  and  -t. 
From  equation  (i)  we  get  the  line  joining  /  and  -/  as  the  determinant 


(17) 


X  n         4  X 1 


4f,{a,t'+2bit  +  c,y  ,   (aj  f  +  2  62  /  +  c^ )' 
d^f  ,{aif  —  2  Z),  /  +  cy  ,   (a^  t'  —  2b^t  +  c^Y 


-   o. 


Expanding    and    removing    extraneous    factors    and    placing     /'=Ci/a, 
we  have 

(18)  L(oi,)  =  [ai'biCi'  +  a'b^c^ — la^b^cfii — 2a^a^iC'  -\-2  a^aJbiC^Ci 

+  4  a,  61  62*  c,  —  20,  61'  62  Cj  —  2  ^2  ^1'  ^2  c^  Xq 
+  [  2  Z)2  («!  C2  +  a^  c,)]  Xi  —  4  a^  bi  c^  X2  =  o. 

If  t'  =  c-i/a-i  we  have 

(19)  L(o.2)  -  [ — a^b^c^ — a^b-fi^*  ^2a.^b^c^c.,+2a^a^yC^—2a^a^.f^Ci 
—   4   02  ^1  *^2  C2  +  2  a,  6,  62*  C2  +  2  aj  &,  b^  c,  ]  x^ 

+  4  aj  &j  C2  «!  —  [  2  &,  (  a,  Cj  +  flj  c, )  ]  acj  =  o. 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic     15 
lit'   =   (5i  Cj  —  ^2  ^1  )/(«^i  ^2  —  02^1)  we  have 

(20)  L(o,3)  =  b^'Xi  —  bi'x2  =  o. 

If  L(o,i),  L,o,2),  L,o,3)  are  on  a  point  the  determinant  of  their  coefficients  must 
vanish.     The  determinant  may  be  written 

bi{alcl+alcl) — 2ai&2Ci(aiC2  +^2^1)  +  2^16  [^.^(ajCi— 6162)  +2bfi2Ci{2afi2 — a^b^^a^c^  +a2Ci,2a,Ci 
-b 2{a\cl  +  alc'^  -V2a.})^C2{a^c.^+a2C^—2a.i)2Cy{ayC2 — bp^—2b})2C2{2a]3^ — a^^,2a2C2,a^C2-Va2C^ 

o  ,         ij   .   *i 

This  expanded  is  readily  seen  to  vanish  which  proves  the  theorem  that  the  six 
lines  on  the  points  of  contact  of  tangents  from  the  meets  of  any  two  double  lines 
form  a  complete  four-point. 

These  six  lines  together  with  the  four  double  lines  form  a  Desargues 
Configuration  B.  That  is  we  have  two  triangles  perspective  from  a  point 
and  having  homologous  sides  meeting  in  three  collinear  points. 

We  shall  now  study  the  four-point  more  in  detail.  The  one  point  ob- 
tained was  that  determined  by  the  fines  L(o,i),  L(o,2)  and  L(o,3)-  Thus  the 
point  is  paired  off  with  the  double  line  o.  In  the  same  way  each  of  the  four 
points  is  paired  with  a  double  line.  Now  there  is  reason  to  believe  from  other 
considerations,  that  these  points  are  in  some  way  related  to  the  Stahl  conic  N, 
which  is  the  locus  of  the  flex  lines  of  cubic  osculants  of  the  rational  quartic. 
We  shall  show  that  the  four  points  are  the  polar  points  of  the  four  double  lines 
as  to  the  conic  N. 

If  the  quartic  is  written 

(21)  X,  =  a,- 1*  +  ^  bi  /3  +  6  Ci  f  +  4  ^,-  f  +  e.- ,  [i  =  I,  2,  3], 
it  is  known  that  N  takes  the  form 

(22)  — 36(  bcx)  (cdx)  +12  {ad  x)  (c  d  x)  +12  {b  e  x)(b  c  x)  +4  (b  d  x)' 

+  {aexY  +  4  {ab  x)  {dex)  —  8  (adx)  (b  ex)  =0, 

where  (b  c  x) 

Cq  Ci  C2 

and  so  on. 


bo 

b. 

b. 

Co 

Ci 

Ci 

Xq 

Xi 

X2 

i6     T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic. 
Taking  the  quartic  as  given  by  (i)  N  is 

(23     )xo'[ — 4|oiii(a2C2+2&/)  I  |62C2(a,c,+2fe,*)  |  +4|62C2(aiC,  +25,")  |  {ai't^c^l 
+  4  I  o,  ft,  (flj  C2  +  2  b^')  1 1  a,  6,  C2'  I  +4  |a,  ft,  ^2  C2 1'  +  I  a,'  ^2'  1' 
+  4  I  a,"  ^2  ^2  II  by  CyCi'  I  —  8  I  a,"  ^2^2  I  I  o,  ft,  C2'  |  ] 

+  Xi'  [i6a2b2'c2]  +  x'  [  i6aift,'Ci]  +  x,  ^2  [  —  i6ft,ft2(a^iC2  +  Oz^i)] 

+  Xo  Xj  [  —  8  6,  c,  I  a,  6,  (02  C2  +  2  62')  I  +  8  a,  6,  I  ^2  C2  ( <ii  c,  +2  6,')  | 
+  8  ft,  c,  I  a,"  &2  <^2  I  —  8  a,  ft,  |  a,  ft,  C2'  |  ] 

+  a^o:*;,  [  8  ft2  Cj  I  a,  ft,  (aj  C2  +  2  ft2')  |  —  ^a^h^  |  ft2  C2  (aiC,  +2ft,")  j 

—  8  biCi  I  a,"  ft2  ^2  I  +  8  ^2  ^2  I  ^1  ^1  ^^2'  I  ]  =  o, 
where  |  c,  ft,  (  Oj  ^2  +2  ft,')  j  =  a,  ft,  (  02  C2  +2  ft2'')  —  a^  b,  (a,  c,  +2  ft,'), 
and  so  on. 

The  coordinates  of   the  point  in  question  are  found  by  getting  the  inter- 
section of  any  two  of  the  three  Hnes,  say  Ljo,,)  and  L(o,3).     We  have  at  once 

Xo=  2  ft,  ftj  (c,  I  a,  ft2  I  —  «!  I  fti  C2  I  ) 

(24)  x,=ft,*  [  2  c,|  a,  ft,(a2C2+2  ftj')  I  +2ft,'ft2|a2Ci|  +a,"c2|ftiC2|  +c,  |a,'ft,c,  |] 
X2=ft2''  I  2  c,  I  a,ft,(a2C2  +  2  ft2')  I  +2  ft,'ft2|a2<^i|  +a,'C2|ft,  C2I  +Ci  |ai"ft,c,|] 

where  the  expressions  within  the  bars  have  the  same  meaning  as  in  (23). 
The  question  now  is  whether  this  point  and  the  double  line  o  are  pole  and  polar 
with  regard  to  N.  To  prove  this  we  only  have  to  find  the  derivative  of  N  as 
to  Xi,  and  then  as  to  X2,  and  see  if  the  two  resulting  lines  pass  through  the 
point  represented  by  (24). 

(25)  Da:,N  s  Xo  [biCil  o,  ft,  (aj  C2  +2  ft2")  I  —  aj  ftz  j  ft2  Cj  (a,  c,  +2  ft,*)  j 

—  ft2  C2  I  a,'  ft2  C2  I  +  a2  ft2  I  fli  fti  C2'  I  ] 
+  Xi  [  4  O2  bi'  C2]  +  X2  [ —  2  a,  ft,  ftj  C2  —  2  ^2  bi  bi  c,]  =  o. 

(26)  D»2N  =  Xo  [  —  ft,  c,  I  a,  ft,  (^2^2  +2  bi')  I  +a,  ft,  |  ft2C2(a,c,  +2ft,')  | 

+  ft,  c,  I  a,"  ftj  C2  I  —  o,  ft,  I  a,  ft,  C2'  I  ] 
+  Xi[ — 20,  ft,ft2C2  —  202^1^2^1]  +  ^2  [  4  a,  fti'c,]  =0. 

Substituting  the  coordinates  of  the  point  given  by  (24)  in  (25)  and  (26)  we 
find  that  each  of  them  is  satisfied,  and  the  theorem  is  proved. 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic       17 

We  have  seen  that  there  is  a  single  infinity  of  four-points  on  the  curve 
for  which  (0,1)  and  (2,3)  are  fixed  diagonal  points,  and  we  now  want  to  find 
the  locus  of  the  third  diagonal  point.  Dr.  J.  R.  Conner  has  suggested  and 
kindly  given  me  the  proof  that  the  projection  of  the  intersection  of  two  cir- 
cular cylinders  touching  and  intersecting  at  right  angles  is  a  general  rational 
quartic.  The  general  rational  plane  quartic  may  be  considered  as  a  projec- 
tion of  a  space  quartic  with  a  node. 

A  quartic  in  space  is  the  intersection  of  two  quadric  cones.  Let  Aj  and 
Bi  be  the  vertices  of  two  quadric  cones  on  the  curve.  Choose  the  plane  at 
infinity  as  a  plane  on  A,  B,.  This  plane  meets  each  cone  in  a  pair  of  lines. 
Take  the  absolute  as  a  conic  touching  the  four  lines  and  apolar  to  the  pair 
of  points  A,  and  B,.  Then  the  curve  is  the  intersection  of  two  circular  cylinders 
touching  and  intersecting  at  right  angles* 

We  shall  denote  this  space  quartic  with  one  node  by  the  symbol  S*,  and 
the  plane  rational  quartic  into  which  S*  projects  by  R''.  Now  consider  S*. 
Take  the  line  normal  to  the  two  cylinders  at  the  node.  The  osculating  planes 
of  the  two  branches  through  the  node  contain  this  line.  There  is  a  single 
infinity  of  planes  on  this  line,  each  of  which  meets  S*  in  two  points  other  than 
the  node.  We  have  thus  a  quadratic  involution.  Its  double  points  will  be 
given  when  the  plane  osculates  one  of  the  branches  at  the  node;  that  is  to  say 
the  nodal  parameters  give  the  double  points  of  the  involution.  That  tells  us 
that  all  these  planes  cut  out  from  S*  quadratics  apolar  to  the  quadratic  giving 
the  node.  Projecting  from  a  point  M  of  space,  the  two  tangent  planes  to  each 
cylinder  from  M  go  into  the  four  double  lines  of  R*,  thus  giving  two  pairs  of 
double  lines.  The  involutions,  which  we  have  previously  discussed,  on  the 
meets  of  pairs  of  double  lines  of  R*  are  cut  out  of  S*  by  the  generators  of  the 
cylinders.  Hence  the  double  points  of  such  an  involution  are  given  by  the 
generators  that  touch  S*.  It  is  then  easy  to  see  that  the  parameters  of  the 
isolated  node  are  apolar  to  the  double  points  of  each  involution;  that  is  the 
Jacobian  of  these  pairs  of  points  gives  the  nodal  parameters.  Also  it  is  here 
obvious  that  the  double  points  of  one  involution  are  a  set  of  the  other,  and  that 
the  node  is  in  both  involutions.  This  shows  again  the  paring  off  of  the  double 
lines  by  choosing  a  node. 

The  involutions  on  the  meets  of  double  lines  of  R*  are  the  projections  of 
points  cut  out  on  S'*  by  the  generators  of  the  cylinder.     We  have  on  S*  a 

*Cf.  Marletta:  Stiidio  geometrico  della  guartiea  golba  razionale.  Annali  di  Ma.  Series  3, 
Vol.  8,  pp.  Ulff. 


1 8     T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

system  of  corners  of  rectangles  cut  out  by  planes  on  A;  and  B,;  these  groups 
of  four  points  on  R*  are  obtained  from  any  pair  of  involutions,  such  as  /,o,,) 
and  /(2,3)-  We  may  remark  in  passing  that  it  is  obvious  that  the  parameters 
of  the  groups  of  four  points  of  which  we  are  speaking  are  a  pencil  of  quartics, — 
precisely  a  syzygetic  pencil,  being  built  on  a  quartic  and  its  Hessian — ,as  it 
is  easily  seen  from  the  space  figure  that  the  pencil  contains  three  perfect 
squares. 

The  diagonals  of  the  rectangles  intersect  in  a  point  whose  locus  is  just 
that  of  the  third  diagonal  point  which  we  started  out  to  find.  It  is  seen  that 
this  locus  is  a  line,  and  it  is  the  normal  to  the  cylinders  at  the  node.  This  line 
projects  into  a  line  in  the  plane  which  passes  through  a  node  and  cuts  out  a 
pair  of  parameters  from  R*  harmonic  to  the  nodal  parameters.  It  is  evident 
there  is  only  one  such  line  for  each  node.  We  thus  get  three  such  lines,  and 
Professor  Morley  has  proved  that  these  three  lines  are  on  a  point.  We  have 
then  the  theorem:  With  each  pairing  of  the  double  lines  of  R*  we  obtain  groups 
of  four-points  on  R*,  which  have  the  meets  of  the  pairs  of  double  lines  as  fixed 
diagonal  points,  and  whose  third  diagonal  point  has  for  a  locus  the  line  on  the 
isolated  node  and  meeting  R*  in  harmonic  pairs. 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic     19 


§  3.     The  Case  with  Three  Flex  Tangents  on  a  Point. 


It  is  well  known  that  the  locus  of  lines  which  meet  a  rational  quartic 
in  sets  of  four  self  apolar  points  is  a  conic,  g^.  In  particular  the  flex  tangents 
are  such  lines;  hence  the  six  flex  tangents  touch  the  gj  conic.  Now  if  gj 
breaks  up,  it  breaks  up  into  two  points.  Then  three  flex  tangents  are  on  one 
point  and  three  are  on  the  other  necessarily. 

First  take  a  quartic  curve  referred  to  two  flex  tangents  and  the  line 
joining  the  two  flexes: 

Xa  =^  at*  +46/' 
(i)  X,  =  46  ^^  +  6c/*+4i/ 

Xi  =  ^  d  t  +  e 

We  now  have  two  flex  tangents  on  the  meet  of  x^  and  x^.     If  a  =  2  6,  and 
e  =  2  dwe  shall  have  a  third  on  that  point,  and  the  curve  takes  the  form: 

Xa  =  2bt*+  ^bt* 

(2)  x^=4bt^+6ct'+^dt 
X2  =  4^d  t  +  2  d 

The  three  flexes  have  the  parameters  o,  00 ,  — •  i. 
The  cubic  giving  them  is  then 

(3)  t'  +t  =0. 

The  flex  tangents  meet  the  curve  again,  and  the  cubic  giving  the  para- 
meters of  these  three  points  is 

(4)  2  /3  +  3  f  —  3  /  —  2  =  o, 

which  is  seen  to  be  the  cubicovariant  of  the  flexes  given  by  (3). 


20     T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

We  wish  to  find  the  cubic  giving  the  parameters  of  the  other  three  flexes. 
The  Jacobian  of  the  two  members  of  the  fundamental  involution  gives  the 
sextic  of  flexes.     Calculating  the  fundamental   involution  of   (2)   we  have 

(5)  2bt''+2ct  +  c  =  o,  and 

(6)  c  i*  +  2  c  t^  +  2  d  t'  =  o. 
The  Jacobian  of  (5)  and  (6)  is 

(7)  2bt^  +  {:ic  +  2b)t^+6ct^  +  i3C  +  2d)t'+2dt=o. 

The  factors  of  (7)  are 

(8)  t'  +  t  =  o  ,  and 

(9)  2  5  ^3  +  3  c  f  +  3  c  ^  +  2  (i  =  o. 

We  shall  speak  of  the  two  points  on  each  of  which  are  three  flex  tangents 
as  the  g2  points,  and  the  line  on  them  as  the  g2  line.  Now  the  six  tangents  from 
one  of  these  points  is  a  cubic  squared,  say  [(«  /)^]^  and  the  six  on  the  other  is 
say  [(&  /  )3  ]^  The  six  tangents  from  any  point  on  the  gj  line  will  be  in  the 
pencil 

[(«/)3]._X»[(M)3]>  =  o,  or 

(10)  [  (a  O'  +  ^  (  M  )M  [  (  «  O'  —  ^  (  M  )M  =  o. 

This  shows  that  the  tangents  all  along  the  gi  line  pair  oflF  into  two  sets  of 
three.     In  our  notation  the  pencil  of  cubics  is 

(11)  2bt^  +  2,ct'+2,ct  +  2d  +  \{t'+t)=o, 

which  is  readily  seen  to  be  a  set  of  apolar  cubics. 

The  equation  of  the  gj  liiie  is  x^  —  X2  —  o.     Hence  the  parameters  of 
the  four  points  in  which  this  line  meets  the  curve  are  given  by 

(12)  bt*  +  2bt^  —  2dt  —  i  =  o. 

At  these  points  two  of  the  six  tangents  to  the  curve  come  together.  The 
pairing  of  the  tangents  must  be  this  tangent  at  the  point  with  two  of  the 
remaining  four,  and  this  same  tangent  with  the  other  two;  or,  the  tangent  at 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic     21 

the  point  taken  twice  with  one  of  the  four,  and  then  the  remaining  three  paired. 
The  latter  is  the  case,  as  is  proved  by  taking  the  Jacobian  of  the  two  members 
of  (11)  and  obtaining  (12).  We  note  too  that  (12)  is  a  self-apolar  quartic 
which  tells  us  again  that  the  pencil  of  cubics  are  apolar,  since  they  have  a  self- 
apolar  quartic  for  Jacobian. 

The  pencil  of  cubics  has  a  unique  apolar  quartic  and  its  Hessian  is  the 
Jacobian  of  the  cubics.     Finding  the  quartic  we  have 

(13)  Q  ^  b't*  +  4bdt^  +  6bdt'  +  ^.bdt  +  d'  =0. 

The  Hessian  of  this  quartic  is  at  once  verified  to  be  (12). 

At  the  four  points  in  which  the  g2  line  meets  the  curve  two  tangents  come 
together  and  we  have  just  seen  that  one  other  tangent  is  paired  with  this  one 
taken  twice.  There  are  then  four  such  tangents.  They  are  given  by  the 
Steinerian  of  Q  to  which  the  system  is  apolar,  for  if  the  Hessian  has  a  root 
that  is  a  double  root  of  the  cubic,  the  other  root  of  the  cubic  is  a  root  of  the 
Steinerian.  The  Steinerian  of  a  quartic  Q  is  known  to  be  gj  Q  +  X  giH  =0, 
when  g2  and  g,  are  the  invariants  of  Q,  and  H  is  its  Hessian.  But  g^  of  our  Q 
is  zero,  so  the  Steinerian  is  the  quartic  over  again. 

We  assert  further  that  Q  is  the  quartic  of  which  the  system  of  cubics  are 
the  first  polars.  Salmon  tells  us  how  to  find  the  quartic  when  the  cubics 
are  given.  It  is  12  H  (J)  +  g^  J,  where  J  is  the  Jacobian  of  the  cubics,  H 
the  Hessian  of  J,  and  ^2  the  self  apolarity  condition  of  J.     But 

(14)  ]^b^^  +  2bt^  —  2dt  —  d=o, 
and  g2  -  o,  and 

(15)  H  (J)  ^  b't*  +  4bdt3  +  6bdt^  +  ^bdt  +  d'  =0. 

This  is  just  Q  over  again. 

Q  is  then  the  quartic  to  which  the  system  of  cubics  is  apolar,  as  well  as  the 
quartic  of  which  they  are  the  first  polars,  and  besides  it  is  its  own  Steinerian,  and 
gives  the  points  of  contact  of  the  four  tangents  which  are  paired  with  the  tangents 
counted  twice  at  the  points  where  the  gj  ^ine  meets  the  curve. 

Consider  now  the  pencil  of  cubics.  To  any  one  of  the  pencil  we  have  a 
definite  corresponding  point  /,  on  the  curve,  viz.  the  point  with  regard  to  which 
the  cubic  is  the  polar  of  Q.  But  paired  with  that  cubic  there  is  a  second  cubic, 


22     T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

and  we  get  a  point  /j  on  the  curve  from  it  in  a  similar  way.  We  have  then  a 
quadratic  involution  set  up  on  the  curve.  Since  the  two  cubics  come  together 
at  the  g2  points,  these  points  correspond  to  the  double  points  of  the  involu- 
tion. We  can  find  the  quadratics  giving  the  double  points  in  the  following 
manner.  The  coefficients  of  the  polarized  form  of  Q  must  be  proportional 
to  the  coefficients  of  the  pencil  of  cubics.  We  can  thus  determine  X  in  terms 
of  /,. 

Polarizing  Q  we  have 

(i6)  (b't^+bd)  t^+3ibdt,+bd)  t'-h2,(bdt^+bd)  t+bdti+d'  =o. 

The  pencil  of  cubics  is 

(17)  2bt^  +  {sc+\)t'+{3c  +  \)t  +  2d=o, 

hence 

(6bd  —  ^b  c)  ti+6b  d  —  ^cd 


(18)  X  = 


b  ti  +  d 


Since  our  base  cubics  are  those  at  the  gj  points,  the  double  points  of  the 
involution  will  be  given  for  X  =  o,  and  X  =  00  ;  that  is  one  is  given  by  the 
numerator  of  (18)  and  the  other  by  the  denominator  of  (18). 

We  now  look  for  this  pair  of  points  on  the  curve.  The  conic  on  the 
flexes  meets  the  curve  again  in  two  other  points  q.  If  the  fundamental  in- 
volution of  the  curve  is  (a  t)*  +  X  (^  /)*  then 

(19)  g  ^  («^)3(aO(M)   =0. 

Using  the  fundamental  involution  as  given  by  (5)  and  (6)  we  get 

(20)  q  =  {b  —  c)t'  —  ct  +  (d  —  c)  =0. 
Now  operating  with  q  on  the  pencil  of  cubics  we  obtain 

(36c  —  6  b  d)  ti  +  3cd  —  6  b  d 

(21)  X  =       ~     . 

bti+  d 


T.  B.  Ashcraft:  Quadratic  Involutions  on  the  Plane  Rational  Quartic.      23 

It  is  to  be  noticed  that  X  in  (21)  is  just  the  negative  of  X  in  (18).  This 
shows  a  sort  of  cross  working  of  the  quartic  of  which  the  cubics  are  the  first 
polars,  and  the  quadratic  g;  that  is  if  C,  is  the  polar  cubic  as  to  /„  and  Cj 
is  the  polar  cubic  as  to  ti,  then  q  operating  on  C,  gives  ^2  and  q  operating  on 
Cj  gives  /j. 

The  double  points  of  the  quadratic  involution  are  the  points  with  regard 
to  which  the  polars  of  the  quartic  are  taken  to  give  the  two  base  cubics  of  the 
pencil,  that  is  the  flex  cubics.  Now  at  these  double  points  we  have  a  tangent 
from  some  point  on  the  52  line;  that  is  each  of  these  belong  to  one  of  the  cubics 
of  the  pencil.     We  seek  the  relation  of  this  cubic  to  the  flex  cubic. 

The  polar  of  (13)  as  to  the  double  point  b  t  +  d  gives  the  flex  cubic 

(22)  t"  +  t  =  o. 

We  can  find  the  cubic  to  which  bt+d  belongs  by  equating  coefficients  in  the 
identity 

(23)  2  b  t^  +  {2,c+-k)t'  +  {2>c+'>^)t+2d^{bt+d)  (a^t'+a.t+a,), 
and  we  get 

(24)  {bt+d){t'+t  +  i)   =0. 

The  last  factor  gives  the  pair  of  tangents  to  which  bt+d  belongs,  and  it  is 
seen  to  be  the  Hessian  of  (22). 

Now  the  cubic  /*  +  /  =  o  which  we  have  considered  is  special  only  geo- 
metrically; it  is  one  of  the  pencil  and  behaves  as  any  other  member  of  the 
pencil.  So  any  two  of  the  three  tangents  given  by  (24)  are  the  Hessian  of  some 
cubic  of  the  pencil.  We  look  for  the  three  cubics  thus  obtained  from  (24). 
The  factored  form  of  (24)  is 

(25)  (bt  +  d)  (t  —  w)  (t  —  w')  =0. 

We  assert  that  the  three  cubics,  of  which  any  two  of  the  three  tangents  given  by 

(25)  are  the  Hessian,  are  just  the  cubics  which  are  the  polars  of  the  quartic  Q  as 
to  the  three  roots  of  (25).     The  polar  of  Q  as  to  /  =  w  is 

(26)  {b'w  +  bd)  t^  —  T)bdw'f--2)bdw't  +  bdw  +  d'  =0. 


24     T.  B.  Ashcraft:  Quadratic  Involutions  on  the  Plane  Rational  Quar tic 

The  Hessian  of  (26)  is 

(27)        (bt  ^-  d)  {t  —  w')  =  o, 

and  our  assertion  is  proved.  That  is  we  have  a  system  of  cubics  which  are  the 
first  polars  of  a  quartic,  such  that  if  the  roots  of  any  one  be,  /,,  /j,  t,,  then  the  cubic 
which  is  the  polar  of  the  quartic  a^  to  any  one  of  the  /'s,  and  is  therefore  one  of  the 
pencil,  has  the  other  two  t's  for  its  Hessian. 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic     25 


§  4.     The  Case  with  Three  Double  Lines  on  a  Point. 


First  of  all  we  shall  find  the  parametric  equation  of  a  quartic  curve  with 
a  syzygetic  point,  that  is  with  three  double  lines  on  a  point.     We  may  choose 
three  points  of  the  curve,  say  o,  00  ,  i.     Let  o  and  00  be  the  points  of  contact 
of  one  double  tangent,  say  «,  =  t". 
Let  «o  be  a  double  tangent  with  points  of  contact  i  and  a ;  then 

X,  =  (t—lYit—a)'. 

If  there  is  another  double  tangent  on  the  meet  of  x^  and  Xi,  it  is  of  the  form 
a;o  +  ^  ^1  =  o,  and  we  find  that  «  =  i,  and  X  =  4. 

The  fourth  double  tangent  will  be  written  generally,  and  the  equation  of  the 
curve  is 

x,=^{t  —  iy{t  +  iy 

(1)  x,=t' 

Xt  =  (/*— 5,/  +  s^y. 

By  transformation  we  can  write  the  curve  in  the  better  form 

x^  =  t*  +  I 

(2)  X.  =  6  /'  2(5, '—  i) 

X2  =  ^fi  +  ^Sjt  +  m.         [m  = ] 


The  three  double  lines  on  a  point  are 

X,  =  O,     3  ATo  +  »,  =  o,  3  »o  —  »,  =  o. 
The  Hessian  of  the  three  double  lines  is 


26     T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

(3)  3  Xo'  +  X,'  =  o. 

The  cubicovariant  of  the  three  double  lines  is 

(4)  x^  {x^'  —  Xo')  =  o. 
If  we  cut   (2)   by  any  line 

(5)  (^  x)  =  $^x  +  ^,Xi  +  eja;^  =  o, 
we  have  a  quartic  in  t 

(6)  fo  ^*  +4  f  2  t^  +  6  ^ 1 1'  +  /^  ^ 2  Sz  t  +  m  $ ,  +  $ 0  =  o. 

Now  if  we  form  the  two  invariants,  gi  and  g,,  of  (6)  we  get  a  line  conic  and 
a  line  cubic. 

We  shall  prove  that  the  lines  to  the  gi  conic  from  the  syzygetic  point  are  the 
Hessian  pair  of  the  three  double  lines  on  that  point;  and  that  the  three  lines  to 
the  gi  cubic  from  the  syzygetic  point  are  the  cubicovariant  of  the  three  double  lines. 

Writing  the  gi  of  (6)  we  get 

(7)  fo'     +    3f."—  4^2f2"     +    Wf    0^2     =    o. 

Put  62=0  and  we  get  the  tangents  to  g^  from  the  syzygetic  point.  Changing 
the  result  into  points  we  have 

(8)  3  X,'  +  X."  =  o. 
Next  writing  the  g^  of  (6)  we  have 

(9)  f,3+^f^3_f^»f^  +  (5^^  +  l)   f„3f^=.+2  52f,f3»+mfoflf2    =   o. 

Setting  f  2  =  o,  and  changing  the  result  into  points  we  have  as  the  tangents 
from  the  syzygetic  point 

(10)  x^  (  x^"  —  x^')   =-  o, 
and  our  theorem  is  proved. 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic     27 

The  Stahl  conic  N  was  defined  in  section  2  of  this  paper.      The  cubic 
osculant  at  a  point  t  of  the  quartic  (2)  is 

Xo     =     T  /'     +     I 

(11)  X,    =  3i'  +  3  ^  t 

X2  =  t^  +  3  -:  t'  +  3Sjt  +  S2  ■:  +  m. 

Cut  this  cubic  by  two  lines  {u  x)  and  (vx).  Making  them  cut  in  sets  of 
apolar  points  we  get  the  equation 

(12)  :«i  (3^2  — 3 -f')    +  x^is^x'  +  rtfz  —  i)  =  o.      . 

For  a  given  t  this  a  line,  the  line  of  flexes  of  the  cubic  osculant  at  the  point 
T  of  the  quartic.  For  a  varying  x  we  get  the  locus  of  this  line,  which  locus  is 
the  Stahl  conic  N.     Its  parametric  equation  is 

^0  =  3^2  —  3^' 

(13)  $1  =  S2  x'  +  m  X  —  I.    . 

f  2  =  o 

Since  fj  =  o  all  the  flex  lines  of  the  cubic  osculants  pass  through  that 
point,  which  is  the  syzygetic  point.  That  is  N  is  the  syzygetic  point  counted 
twice. 

The  cubic  osculants  at  the  points  of  contact  of  double  tangents  have 
interesting  properties.  Consider  the  double  tangent  with  points  of  contact 
I  and  —  I.     The  cubic  osculant  at  t  =  i  is 

x^  =  t^  +  I 

(14)  ^1  =  3  ^'  +  3  ^ 

X2  =  t^  +  3f  +  3S2t  -{■  S2  +  m. 

This  curve  passes  through  N  and  of  course  has  a  flex  there.  The  flex 
tangent  is 

(15)  Xa  +  x^  =  o.  • 
The  cubic  osculant  at   t  =  —  i   is 

Xo   =  —  t^   +   I 

(16)  x,  =^  St'  +  3t 

X2  =  t^  —  3  /*  +3^2^  —  ^2  +  m 


28     T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic 

It  also  passes  through  N,  and  has  then  the  same  flex  tangent  as  (14).  Hence 
the  theorem:  When  three  double  lines  are  on  a  point  the  cubic  osculants  at  the 
two  points  of  contact  of  any  one  of  these  three  double  lines  pass  through  the  syzy- 
getic  point,  each  having  ajlex  there,  and  both  have  the  same  flex  tangent. 

The  three  such  tangents  are  also  the  three  tangents  tog,  from  the  syzygetic 
point. 

We  have  seen  that  a  line  on  the  points  of  contact  of  the  two  tangents  from 
the  meet  of  any  two  double  tangents  passes  through  the  meet  of  the  other  two 
double  tangents.  In  the  syzygetic  case  the  six  such  lines  are  on  the  syzygetic 
point,  three  of  them  being  the  double  lines  themselves.  The  equations  of 
the  other  three  may  be  found  by  use  of  the  catalectic  sets  of  the  fundamental 
involution. 

Again  the  Jacobians  of  the  factors  of  the  fundamental  involution  give 
the  nodes,  and  thus  we  can  get  the  equation  of  the  lines  to  the  nodes  from  the 
syzygetic    point. 

We  have  then  four  sets  of  three  lines  on  the  syzygetic  point,  viz.. 
The  three  double  lines 


Xi  =  0 

W 

I                      2x0  +  Xi  =  0     , 

(B) 

3x0        X,  =  0     . 

(C) 

The  lines  to  ^3 

Xo  =  0 

(A) 

II                    Xi  — Xo  =  0      , 

(B) 

:»,  +  :Vo  =  0     . 

(C) 

The  lines  to  the  nodes 

6  S2X0  +     {Si'  +  I  )  :!C,  =  o  ,  (A) 

III  3{s,  —  iyx,  —  [2s^'  +  (s,  —  iy]x,=o,(B) 

3  (s,  +  i)'  Xo  —  [2  5,"  —  (  52  +  I  )=■  ]x,  =0.  (C) 

Lines  on  the  points  of   contact    of   tangents   from    the  meets    of    the  three 
double  lines  on  the  syzygetic  point  with  the  fourth  double  line 

es^Xa—    (52*   +   I  )  :JC,    =  O  ,  (A) 

IV  2>s,'x,  —  [2{s^  —  i,y+s,']x^=o,  (B) 

3Si'Xa—[2{S2   +   iy  —  Si']Xt=0.  (C) 


T.  B.  Ashcraft:    Quadratic  Involutions  on  the  Plane  Rational  Quartic     29 

We  have  already  seen  that  the  second  set  is  the  cubicovariant  of  the  first 
set.  We  say  furthermore  that  the  lines  marked  (A)  are  harmonic,  because  it  is 
readily  seen  that  they  are  of  the  form 

{<x  x)  =0,  (Prv)   =  o,(  '^  x)   +  X  (i^  x)  =o,(ax)  —  X(^x)  =0. 

The  same  is  true  of  those  marked  (B),  and  of  those  marked  (C). 

In  conclusion  we  may  add  that  the  study  of  the  syzygetic  case  of  the 
rational  quartic  is  the  same  as  that.of  a  conic  and  four  points  such  that  conies 
on  them  cut  out  a  syzygetic  pencil  from  the  given  conic.  For  if  we  call  the 
nodes  do,di,  dj,  and  the  syzygetic  point  N,  and  then  invert  the  quartic  into  a 
conic  a  by  the  transformation  j,-  =  i/xf,  the  four  points  all  behave  alike. 
The  double  lines  become  conies  which  touch  the  conic  a  twice.  Three  of 
these  meet  at  N.  The  conic  on  d^,  di,  d^^,  and  bitangent  to  a  would  corres- 
pond to  the  fourth  double  line.  We  have  four  points  dg,  d!„  d-^,  N,  such  that 
conies  on  them  cut  out  a  syzygetic  pencil  from  a,  that  is  such  that  three  con- 
ies can  be  drawn  on  them  bitangent  to  a. 

Johns  Hopkins  University,  April  1911. 


VITA 

Thomas  Bryce  Ashcraft  was  born  at  Marshville,  North  CaroHna  on 
November  27,  1882.  He  was  prepared  for  college  at  the  Wingate  High  School. 
He  was  graduated  from  Wake  Forest  College  in  1906  with  the  degree  of 
Bachelor  of  Arts.  In  October,  1907,  he  entered  the  Johns  Hopkins  University 
with  Mathematics  as  his  principal  subject,  and  Physics  and  Astronomy  as 
first  and  second  subordinates.  He  held  a  North  Carolina  scholarship  during 
the  years  1907-1911. 

He  wishes  to  th^nk  Doctors  Morley,  Cohen,  Coble,  Hulbert,  Bliss, 
Pfundt,  and  Anderson  for  valuable  instruction  and  encouragement  in  his  uni- 
versity course.  He  also  expresses  his  sincere  gratitude  to  Professor  Morley 
for  helpful  suggestions  and  constant  inspiration  in  the  preparation  of  this 
dissertation. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
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